It is presumably well known that, e.g., 𝜗n(a π,e^-(b π)) = algebraic Γ(1/4)/π^(3/4), a,b ∋ Q, so the only problem is to find the algebraic. Instead of PSLQing a big polynomial of unknown degree to unknown precision and then struggling to identify and simplify its relevant root, we can exploit the (Ramanujan's?) observation that these algebraics are often of a peculiar form, i.e. products of rational powers of "oligonomials". For example, targeting 𝜗₁(π/6,e^-(π/4)): In[146]:= Log[{(\[Pi]^(3/4) EllipticTheta[1, \[Pi]/6, E^(-\[Pi]/4)])/Gamma[1/4], 2, 3, Sqrt[2] - 1, Sqrt[3] - 1, Sqrt[3] - Sqrt[2], Sqrt[6] - 1, 2^(1/4) - 1, 3^(1/4) - 1, 3^(1/4) - 2^(1/4), 6^(1/4) - 1, Sqrt[3] - Sqrt[2] + 2, Sqrt[2] - 3^(1/4)}]; In[147]:= FindIntegerNullVector[%].% Out[147]= 17 Log[2] - 6 Log[-1 + Sqrt[2]] - 3 Log[Sqrt[2] - 3^(1/4)] - 12 Log[-1 + 3^(1/4)] + 11 Log[-1 + Sqrt[3]] + 6 Log[-Sqrt[2] + Sqrt[3]] + 24 Log[(\[Pi]^(3/4) EllipticTheta[1, \[Pi]/6, E^(-\[Pi]/4)])/ Gamma[1/4]] In[152]:= Solve[0 == ReplacePart[%147, {-1, 2} -> x]] Out[152]= {{x -> 1/24 (-17 Log[2] + 6 Log[-1 + Sqrt[2]] + 3 Log[Sqrt[2] - 3^(1/4)] + 12 Log[-1 + 3^(1/4)] - 11 Log[-1 + Sqrt[3]] - 6 Log[-Sqrt[2] + Sqrt[3]])}} So the answer is In[153]:= %147[[-1, 2, 1]] == Simplify[E^x /. %[[1]]] Out[153]= (\[Pi]^(3/4) EllipticTheta[1, \[Pi]/6, E^(-\[Pi]/4)])/ Gamma[1/4] == ((-1 + Sqrt[2])^(1/4) (Sqrt[2] - 3^(1/4))^(1/8) Sqrt[-1 + 3^(1/4)])/( 2^(17/24) (-1 + Sqrt[3])^(11/24) (-Sqrt[2] + Sqrt[3])^(1/4)) In[154]:= N[List @@ %, 22] Out[154]= {0.3173112547318581725643, 0.3173112547318581725643} So we just collect a bunch of "logarithmically independent oligonomials". Unfortunately, when q=e^-(surd π), things get a bit hairier: Out[155]= EllipticTheta[1, \[Pi]/3, E^(-((Sqrt[19] \[Pi])/3))] == (1/( 4 2^(17/76) Sqrt[ 3] \[Pi]))(-2 + (-5 + Sqrt[57])/(1 + 3 Sqrt[57])^(1/3) + ( 4 (1 + 3 Sqrt[57])^(1/3))/(-5 + Sqrt[57])) (( Gamma[1/38] Gamma[5/38] Gamma[9/38] Gamma[11/38] Gamma[7/19] Gamma[ 17/38] Gamma[23/38] Gamma[25/38])/( 19 Gamma[13/38] Gamma[8/19]))^(1/4) In[156]:= N[List @@ %, 22] Out[156]= {0.5533022123819674888146, 0.5533022123819674888146} --rwg